Math101 · Independent Ontario learning library
Math101learn.math101.caIntroduction to Functions
A function assigns exactly one output to each allowed input and provides a precise language for relationships, graphs, tables, and models.
A function is a relationship in which every allowed input has exactly one output.
The machine idea
Imagine a machine that accepts an input, applies one dependable rule, and produces an output. If the rule is “triple, then subtract two,” input $5$ produces $13$.
Different inputs may share an output, but one input cannot produce two different outputs within the same function. That single-output rule is the defining feature.
Function notation
$f(x)$ means “the output of function $f$ at input $x$.” It does not mean $f$ multiplied by $x$.
If $f(x)=3x-2$, then
The letter naming a function can change. $g(t)$ may describe height at time $t$; notation helps identify both the rule and input variable.
Domain and range
The domain is the set of allowed inputs. The range is the set of outputs actually produced.
Context and algebra can restrict the domain. For $f(x)=1/(x-4)$, $x=4$ is excluded because it would divide by zero. In a model of ticket sales, negative or fractional ticket counts may be excluded even if the formula accepts them algebraically.
Tables and ordered pairs
A function may be represented by a table or set of ordered pairs. The pairs
form a function because each first coordinate appears with only one second coordinate. Adding $(2,7)$ would break the function rule because input $2$ would have two outputs.
Graphs and the vertical-line test
A graph represents a function when every vertical line intersects it at most once. A vertical line fixes one input $x$; two intersections would mean that input has two outputs.
A circle fails the test. A nonvertical line, parabola, and many curves pass. The test concerns output uniqueness, not whether the graph is straight.
Evaluating and solving
Evaluating supplies an input and asks for the output. Solving $f(x)=10$ supplies an output and asks which inputs produce it.
Comparing representations
An equation shows an exact rule. A table highlights selected values. A graph reveals shape, intercepts, intervals, and trends. A verbal description provides context. Strong mathematical reasoning moves between these forms rather than treating them as separate topics.
Linear and nonlinear functions
A linear function has constant rate of change and graphs as a nonvertical line. A quadratic function does not have constant first differences; its graph is a parabola. Both are functions because every input has one output.
Common mistakes
Reading $f(x)$ as multiplication. It names an output.
Reversing domain and range. Domain is input; range is output.
Believing repeated outputs are forbidden. Repeated inputs with different outputs are forbidden.
Applying the horizontal-line test. The vertical-line test identifies functions; horizontal lines test one-to-one behaviour.
Quick self-check
- Does each input have exactly one output?
- What restrictions belong to the domain?
- Am I evaluating an input or solving for an input?
- Which representation best reveals the feature I need?
Related topics
Explore the idea
Function transformation
Change one quantity at a time and connect what moves to Introduction to Functions.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
If f(x) = x² + 1, find f(−3).
- f(−3) = (−3)² + 1
- = 9 + 1
- = 10
End of lesson
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